Re: JSH: SF: Finally, surrogate factoring

[Rick Decker]
...
So completing the square w.r.t y first yields

(2*y + 10*z + 5*f_1 - f_2)^2 = (4*z + 3*f_1 - f_2)^2 + 4*T

Completing the square w.r.t. z first yields

(42*z + 10*y - 3*f_2 + 19*f_1)^2 = (4*y + 3*f_2 - 5*f_1)^2 + 84*T

and rewriting both these as differences of squares yields the same
(useless) factorization of T:

T = (y + 3*z + f_1)(y + 7*z + 4*f_1 - f_2)

Macsyma agrees ;-)

and it's not hard to verify that

g_1 = y + 3*z + f_1

g_2 = y + 7*z + 4*f_1 - f_2

Given that James started with (although it was obscured by the
presentation):

f_1 = w + x - 2*z
f_2 = w + 3*x + 2*y + 2*z
g_1 = w + x + y + z
g_2 = 3*w + x - y - 3*z

that's immediate. So, after small mountains of tedious manipulation, we get
back a minor respelling of the initial assumptions.

What I don't understand is how anything other than that outcome could be
_hoped_ for here. No amount of rearranging and cross-substituting the
initial equations (whatever they may be) is going to yield new information,
and there's never a step that even requires the quantities to be integers
(as opposed to, e.g., arbitrary complex numbers). How can someone imagine
that insight into integer factorization could result from this insight-less
symbol-pushing?

As usual, I couldn't make sense of his original writeup before you showed
the correct result of completing the square wrt y first, at which point I
could work backward from that to deduce what you thought James was trying to
say. Also as usual, you got that right. Therefore :-) you must also know
why he thinks this kind of approach _could_ yield something useful.

Or is this another case where you know he's right, and are keeping silent
about which initial equations actually do work just to protect your career?
Clever, if so ;-)


.

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